- Email: [email protected]
A decision support tool for procurement planning process under uncertainty
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
A decision support tool for procurement planning process under uncertainty R. Guillaume*,**, C. Thierry*, B. Grabot** *Université de Toulouse IRIT/UTM, 5, Allées A. Machado, F-31058 Toulouse Cedex 1 France (e-mail: [email protected] , [email protected]). **Université de Toulouse, INPT/LGP/ENIT 47 Av. d'Azereix, BP 1629, F-65016 Tarbes Cedex France (e-mail: [email protected]) Abstract: This communication presents a method to support the customer in the choice of a procurement plan when the gross requirements are ill-known, in a context of collaboration with the supplier. A general model of imperfect parameter representation is suggested, imperfection gathering uncertainty (through various scenarios) and imprecision (through quantities and dates expressed by possibility distribution). A method to compute the possible quantities required to satisfy the gross requirements under the supplier delivering constraints is proposed. From this value, a set of possible supplied quantities is computed to support the decision making of the customer. The decision maker then evaluates the procurement plan with the possible evolution of the inventory. Keywords: supply chain, uncertainty, collaboration, theory of possibility. build the set of all the possible procurement plans. This method is composed of three steps:
1. INTRODUCTION Enterprises are nowadays more and more tightly integrated in supply chains whereas the globalization of the market and the reduction of the product life cycles increases the uncertainty on the customer's demand. Thus, taking into account uncertainty in supply chains is becoming a major issue. If no historical data allowing to build a stochastic model is available, uncertainty is often taken into account in the literature by possibility theory and fuzzy set theory (Dubois and Prade, 1988) (see for instance (Peidro et al., 2009), (Lan et al., 2009), (Aliev et al., 2007)). These methods have for instance been used to model the imprecision on the coefficients of a cost function (Demirli and Yimer, 2006) or the weights of constraints (Mula, 2007). Moreover, in an uncertain environment, a collaborative process between actors of the supply chain (customer, supplier) is a powerful tool to reduce the risk of the bullwhip effect (Galasso et al., 2006). One of the references of collaborative processes in the industry is CPFR (Collaborative Planning forecasting replenishment (Ireland and Crum, 2005)). Within this framework, data such as inventory levels, forecast, etc. are exchanged between the supplier and the customer. In order to take into account more explicitly the uncertainty related to the demand, a method is suggested in this communication aiming at supporting the decision maker (the customer) in building a procurement plan for the supplier. This decision is made under uncertainty on the gross requirements, in collaborative context (the supplier sends to the customer its delivering capacity constraints). To support the decision maker in finding a feasible procurement plan, we
978-3-902661-93-7/11/$20.00 © 2011 IFAC
◊
1) the computation of the required inventory for each period,
◊
2) the computation of the possible procured quantity under the delivering constraints.
◊
3) the evaluation of the proposed procurement plan according to the possible evaluations of the inventory.
This paper is organized as follows: the first section presents a model of imperfect gross requirement. In the second one, a method to compute the required inventory is suggested while in the third section, a method to compute the possible procured quantity for a period is described, together with the framework of a decision support process aiming at building a procurement plan. In the last section, this decision support process is illustrated on an example. 2. IMPRECISE AND UNCERTAIN GROSS REQUIREMENTS Our main hypothesis is that the customer has knowledge on gross requirements including imperfections (uncertainty and imprecision), often empirically modelled through expertise. So, the gross requirements can be given by the customer as a set of scenarios, with an imprecision on the value (e.g. "Not more than twenty parts, but certainly between 15 and 20"). Possibility theory is then used to model the uncertainty and imprecision on the demand in a more analytical way. Without loss of generality, uncertainty is represented as often by a possibility level (definition 1) and the imprecision by trapezoidal possibility distributions (definition 2 and figure 1). All the scenarios are synthesized in a graph (definition 3,
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10.3182/20110828-6-IT-1002.01342
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
figure 2) where a path stands for a possible scenario. The method to compute this graph of gross requirement is developed in (Guillaume et al., 2010). Definition 1: A possibility level denotes to what extent the occurrence of event e ∈ E if possible. The possibility level t is an upper bound of the probability that the event e will
To represent a fuzzy number, an interval is used where the extreme values are gradual numbers. This representation enables to apply the interval analysis without using the α-cuts (Fortin et al., 2008). We also use a set of two linear gradual numbers to represent the trapezoidal possibility distribution.
~
Figure 2 illustrates the model for a fuzzy number A . The gradual number is represented by two
~−
appear Pr(e) ≤ Π (e) and ∃e Π (e) = 1 . Definition 2: A possibility distribution πv of v quantifies the plausibility of the information v. πv is a function of S in L such
~−
~−
values A = [ A (0); A represented by 4 values.
as ∀s ∈ S , π v ( s ) ∈ L and ∃s π v ( s ) = 1 with v denoting an ill-
possibility
known value in S, and L the scale of plausibility ([0,1] for the theory of possibility) (Dubois and Prade, 2006).
1
(1)] , so a fuzzy number is
~ A−
~ A+
Possibility
x
1
Fig. 3. Fuzzy number
~ ~ ~ A = [ A− ; A+ ]
Quantity S
The data are denoted as follows: Fig. 1. Trapezoidal distribution of possibility
t: period with t ∈[1,T]
Definition 3: The graph of the gross requirements is a directed arc weighted layer graph G =< V , E > with quantities on node defined as:
V = V1 ∪ ... ∪ VT with T the number of layers ~ ; v~ ;..; v~ } with such as Vi ∩ V j = φ where V = {v 1 2 n ~ v trapezoidal possibility distribution.
Node set
ct: index of fuzzy gross required quantity on period t with ct∈[1,Ct]
~ ~ ~ Grt ,ct = [Gr − t ,ct ; Gr + t ,ct ] : fuzzy gross required quantity on node (t, ct)
wct ,ct+1 ,t : possibility level of the arc linking the node (t, ct)
i
Arc set E
= (v, w) such as v ∈ Vi & w ∈ Vi +1 and the
weights of arc are defined by a matrix with
M = (Π v , w )( v , w)∈E
max (Π v , w , v ∈ Vi & w ∈ Vi +1 ) = 1 (consequence of
with the node (t+1, ct+1)
Lt = [lt− ; lt+ ] : set of possible quantity of deliveries lt on period t 3. COMPUTATION OF THE REQUIRED INVENTORY
possibility theory).
The first step, which consists in computing the required inventory (definition 4), is presented in this section. Start
Definition 4: The required inventory is the quantity required in the inventory to fulfil the gross quantities required under supplier delivering constraints.
End
This method uses an iterative algorithm, from period T to 1. The dependant variables are denoted: Period 1
~ * ~ *− ~ *+ I rt = [ I rt ; I rt ] : fuzzy inventory required for period t
Period 2
Scenario 1 1
which satisfies all the possible gross requirements,
~ ~ − ~ + I r* t = [ I r* t ; I r* t ] : fuzzy inventory required for period t
Scenario 2
which satisfies one of the possible gross requirement,
Scenario 3
Fig. 2. Example of graph for a horizon of 2 period and 3 scenarios 1591
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
~ * ~ *− ~ *+ I rt , ct = [ I rt , ct ; I rt , ct ] : fuzzy inventory required for node (t,ct) which satisfies all the possible gross requirements,
~ ~ − ~ + I r* t , ct = [ I r* t , ct ; I r* t , ct ] : fuzzy inventory required for node (t,ct) which guarantee the respect of one or more possible gross requirements. When the gross requirements are imperfect (uncertain and imprecise), the required inventory is imperfect too. To deal with imprecision, each required quantity becomes a possibility distribution. To deal with uncertainty, only the two extremes cases are computed: the required inventory which satisfies all the possible gross requirements and the required inventory which satisfies one of the possible gross requirements.
To back-propagate the maximum (minimum) required inventory level and keep the representation of the possibility distribution by 4 values (figure 4), an approximate method is used. This method maximises (minimises) two gradual numbers by a linear function which guarantees that all the values of the exact solution are lower (upper) than the approximate solution (for the minimum: equations 4 to 6 and figure 5). In fact, the maximum (minimum) between two values has in the worst case one intersection point (figure 4).
~ A− ~ B− x
The operator used to back-propagate the required inventory depends on the type of required inventory: •
minimum: used for the inventory which satisfies one of the possible gross requirements,
•
maximum: used for the inventory which satisfies all the possible gross requirements
x
Exact result
Approximate result
Fig. 4. Exact/Approximation of minimum
(
)
~ *+ ~ *+ I rt (0) = max I rt ,ct (0) (4) wct ,ct +1 ,t ≠ 0
It is considered that the required inventory for period T+1 is equal to 0. To compute a required inventory (for example: which satisfy all possible gross requirements I~r * ), the
1 − wct ,ct +1 ,t ~ * + ~ *+ ~ *+ ~ *+ × I rt ,ct (1) − I rt (0) I rt ,ct (1) ⇐ I rt ,ct (1) + (5) wct ,ct +1 ,t
optimization problem (equations 1 to 3) is solved for each node (t,ct) (from (T, CT) to (1,1)) and for the four values of ~ *− *− the required inventory (for example: ~ I rt ,c (0) ; I rt ,c (1) ;
∀wct ,ct +1 ,t 0 < wct ,ct +1 ,t < 1
~ *+ I rt , ct (1) ; I~rt ,c *+ (0) ) .
Figure 5 illustrates the equations defining the minimum
t ,ct
t
(
(
)
)
~ *+ ~ + I rt (1) = max I rt*,ct (1) (6) wct , ct +1 ,t ≠ 0
t
t
between to gradual numbers
~ minimize ( I r * t , ct ) (1) s.t.
~ A−
1 0.6
~ ~ ~ I r * t , ct − Grt , ct + l t+ ≥ I r * t +1 (2) ~ I r * t , ct ≥ 0 (3)
~ ~ A − and B − .
~ B− 1 3
7
~ A−
1
~ B−
Equations (7 & 8) 13 x
1 3
11 13
x
Minimum
The same method is applied for the inventory which satisfies one of the possible gross requirements (with I* becoming I*).
1
~
The extreme required inventory (optimist = I r * − t ,ct ; pessimist
~
= I r * + t ,ct ) corresponds to the extreme bound of the set of the
1
~ gross required quantity for the next period ( I r * t +1 ) and the
11
x
~
extreme bound of set the gross requirement ( Grt ,ct ), so we ~ + ~ ~ ~ ~ have I r * − t ,ct for Gr − t , ct and I r *− t +1 ; and I r * + t ,ct for Gr t ,ct
Fig. 5. Illustration of the approximation of the minimum
~
4. CHOOSE A PROCUREMENT PLAN
and I r *+ t +1 . The required inventory is the minimum inventory needed to satisfy a gross requirement, so the minimum required inventory respecting the delivery constrains (whatever the quantity delivered by the supplier is) corresponds to the +
maximum possible quantity procured: l t .
The second and the third steps of the method, respectively the computation of the possible procured quantity and the evaluation of the possible inventory level for a given procured quantity, are described in this section. These two steps are performed consecutively from the period 1 to T. The dependant variables are denoted:
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
~ * ~ *− ~ *+ Pt , ct = [ Pt , ct ; Pt , ct ] : fuzzy procurement quantity for
~ ~ Pt * = max ( Pt ,*ct ) (10) ct
node (t,ct) which satisfies all the possible gross requirements,
~ ~ − ~ + P*t , ct = [ P*t , ct ; P*t , ct ] : fuzzy procurement quantity for node (t,ct) which satisfies one of the possible gross requirements,
~* ~* − ~* + Pt = [ Pt ; Pt ] : fuzzy procurement quantity for period t which satisfies all the possible gross requirements,
~ ~− ~+ P*t = [ P*t ; P*t ] : fuzzy procurement quantity for period t which satisfies one of the possible gross requirement,
~ I t* : fuzzy possible inventory at the end of period t in the case satisfy all the possible gross requirements are satisfied,
~ I*t : fuzzy possible inventory at the end of period t in the
4.2 Computation of fuzzy possible procured quantities From the possible procured quantities, a crisp value is proposed according to a risk level in terms of inventory. For example, the lower bound of the most possible value for the case which satisfies all gross requirements can be proposed as a first suggestion since it is a prudent solution. Then, two resulting inventories are computed (one aiming at satisfying all the possible gross requirements; one aiming at satisfying one of the possible gross requirements), using the minimum operator for the first one, and the maximum operator for second one. The minimum and the maximum are computed like the required inventory (equations 4, 5 and 6). Equations 11 and 12 compute the inventory levels which satisfy all the possible gross requirements (with xt standing for the proposed value by the decision maker).
~ ~ ~ xt + I t *−1 − Grt , ct = I t , ct ~ ~ I t * = min ( I t *,ct ) (12)
case when one of the possible gross requirements is satisfied, xt : value proposed by the decision maker for the period t δ: difference between the required inventory of period t and the possible inventory for the period t. 4.1 Computation of fuzzy possible procured quantities This method is applied for the two extreme cases concerning all the gross requirements or one of the possible gross requirements. The decision maker receives so a fuzzy set for each case. In order to compute the two fuzzy procured quantities, the following optimization model is solved (equations 7 to 9) for the four values (from P~t ,c *− (0) to t ~ *+ Pt , c t (0) ).
ct
Then, the fuzzy possible procured quantities are computed for the next period. From the first proposition of procurement plan (x1 to xT), the decision maker analyses the evolution of the inventory and modifies the procurement plan until it gives him satisfaction. 5. EXAMPLE In this section, we illustrate on an example the method of computation of the procurement plan under uncertainties. 5.1 Data
minimize ( δ ) (7)
The considered planning horizon is 2 periods and the inventory at period 0 is equal to 0. The imperfect gross requirements are represented in Figure 6 and Table 1, and the delivering constraints in Table 2.
s.t.
~
∀ct (11)
~
~
~
δ = Pt ,c *− (0) + I t *−1− (0) − Grt −,c (0) − I rt − (0) (8) t
t
~ *− lt− ≤ Pt , ct (0) ≤ lt+ (9)
ϭ
ϭ
The required inventory for period 0 is different from the real inventory at the beginning of period 0, which is known. So we are looking for the procured quantity which satisfies the required inventory for the next period (period 1). We associate a possibility level to the possible delivering quantity, corresponding to the possibility level that the quantity will be required. We compute then the fuzzy quantity delivered for the period. In the case of the procurement plan which satisfies all the possible gross requirements, we merge the nodes with the max operator, in the other case with the min operator (equation 10: procurement plan which satisfies all the possible gross requirements)
^ƚĂƌƚ
ϭ ~ Gr2,1
~ Gr1,1
Ϭ͘ϴ
ŶĚ ϭ
~ Gr2, 2
Fig. 6. Imperfect gross requirement Table 1. Gross requirement t, ct
~ Grt −,ct (0)
~ Grt −, ct (1)
~ Grt +, ct (1)
~ Grt +, ct (0)
2,1
12
17
17
25
2,2
15
16
20
22
1,1
6
7
7
8
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Table 2. Delivery constraints Period 1
1 0.8
Period 2
[l ; l ] = [5;17] [l ; l ] = [5;15]
Delivery quantity
1 1
+ 1
1 2
+ 2
0 1
2
1 0.8
From these data, the required inventory is computed.
0 2
3.75
10
5.2 Computation of the required inventory 0
It is considered that the required inventory of period 3 is (0;0;0;0). Then, the required inventory for the node 1 of period 2 is computed as follows. For the last period, the required inventory which satisfies all the gross requirements is equal to the required inventory which satisfies one of the possible gross requirements
~
( Ir
*
2 ,1
~ ~ ~ = I r *2,1 & I r * 2, 2 = I r *2, 2 ).
2
5
7 10
Fig. 7. Maximum Required inventory of period 1 In the same way, but using the min operator, we compute the required inventory which satisfies one of the possible gross requirements (Figure 8). The result of the computation of the required inventory is illustrated in Figure 9.
The lower bound is then computed:
1 0.8
~ minimize ( I r * − 2,1 ) s.t.
~ ~ ~ I r * − 2,1 − Gr − 2,1 + l 2+ ≥ I r * − 3 ~ I r * − 2,1 ≥ 0
0 1
2
1 0.8
In this case, the equation (5) becomes:
0 1.25 2
~ ~ I r * − 2,1 (0) − 12 + 15 ≥ 0 ⇒ I r * − 2,1 (0) = 0 ~ ~ I r * − 2,1 (1) − 17 + 15 ≥ 0 ⇒ I r * − 2,1 (1) = 2
0
The upper bound is:
2
5
7
7 10
Fig. 8. Minimum Required inventory of period 1
~ ~ I r * + 2,1 (0) − 25 + 15 ≥ 0 ⇒ I r * + 2,1 (0) = 10 ~ ~ I r * + 2,1 (1) − 19 + 15 ≥ 0 ⇒ I r * + 2,1 (1) = 2 ~ ~ I r * 2,1 = I r *2,1 = (0;2;2;10) and in the same way, the ~* ~ node (2,2) is computed : I r 2 , 2 = I r *2 , 2 = (0;1;5;7) .
0 0 2
So,
3.75
0 1.25 2
From these results, the required inventory of the node 1 of the period 1 is computed. The back-propagation of the maximum required by the inventory of the period 2 is used for the case which satisfy all the gross requirements (otherwise, the minimum). We apply the equation (4), then the equation (5) and the equation (6). Figure 7 illustrates the calculation.
(
~ *+ ~ *+ ~ *+ I r2 (0) = max I r2 ,1 (0); I r2 ,1 (0)
)
= max(10;7 ) = 10 1 − 0.8 ~ *+ I r2, 2 (1) = 5 + × (5 − 10) = 3.75 0.8
(
~ *+ ~ *+ ~ *+ I r2 (1) = max I r2,1 (1); I r2,1 (1) = max(2; 3.75) = 3.75
)
2
10
10
7 0 1
5
7
Fig. 9. Required inventory 5.3 Computation of fuzzy possible procured quantities First, we compute the fuzzy procurement plan which satisfies all the gross requirements. We apply the equations 11, 12 and 13 at the period 1:
~
δ = P1,1*− (0) + 0 − 6 − 0 ~ *− 5 ≤ P1,1 (0) ≤ 17 ~ *− ~* − so P1,1 (0) = P1 (0) = 6 1594
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
In the same way, we have:
~* + P1 (0) = 17 .
~* − ~* + P1 (1) = 9 P1 (1) = 10.75
5.4 Computation of fuzzy possible procured quantities A prudent solution is proposed: the lower bound of the most possible value for the case which satisfies all gross requirements (value=9). On the left of Figure 10, a set of fuzzy procured quantities is proposed. On the right, the inventory level resulting from the proposition is represented, together with the required inventory level for the next period. Then, the decision maker can modify the first crisp value and the right side to evaluate the new proposition. In the example, he proposes the quantity 11, which leads to a less risked solution. Fuzzy procurement
Inventory level
All GR
6
9 10.75
17
2
3.75
10
1 GR
6 8 9 Proposition 1 :9
15
1.25 2
7
Proposition 2 :11 Inventory = (1 ;2 ;2 ;3) Inventory = (3;4 ;4 ;5)
Fig. 10. Dashboard of the decision maker for period 1 6. CONCLUSIONS
7. REFERENCES Ireland, R. and Crum, C. (2005). Supply chain Collaboration: how to implement CPFR® and other best collaborative practices, Integrated Business Management series and APICS Dubois, D. and Prade, H. (1988). Possibility Theory, Plenum Press New York. Aliev, R.A., Fazlollahi, B., Guirimov B.G. and Aliev, R.R. (2007). Fuzzy-genetic approach to aggregate production– distribution planning in supply chain management. Information Sciences, 177 (20), 4241–4255. Guillaume, R., Thierry, C. and Grabot, B. (2010). Integration of ill-known requirements with dependencies into a gross requirement plan, 8ème ENIM IFAC Conférence Internationale de Modélisation et Simulation, May 10-12 Hammamet, Tunisia. Galasso, F., Mercé, C. and Grabot, B. (2009). Decision Support for Supply Chain Planning under uncertainty. International Journal of Systems Science, 39 (7), 667675. Dubois, D. and Prade, H. (2006). Représentations formelles de l’incertain et de l’imprécis. In : Hermès – Lavoisier, eds. Concepts et méthodes pour l’aide à la décision 1, 11-165. Mula, J., Polera, R. and Garcia-Sabater, J.P. (2007). Material Requirement Planning with fuzzy constraints and fuzzy coefficients, Fuzzy Sets and Systems, 158 (7), 783 – 793. Peidro, D., et al., (2009). Fuzzy optimization for supply chain planning under supply, demand and process uncertainties. Fuzzy Sets and Systems, 160 (18), 26402657. Lan, Y.F., Liu, Y-K. and Sun, G-J. (2009). Modeling fuzzy multi-period production planning and sourcing problem with credibility service levels, Journal of Computational and Applied Mathematics, 231 (1), 208-221. Fortin, J., Dubois, D. and Fargier, H. (2008). Gradual Numbers and Their Application to Fuzzy Interval Analysis, Fuzzy Systems, IEEE Transactions on , 16(2), pp.388-402, doi: 10.1109/TFUZZ.2006.890680
In this paper is proposed a decision support tool for helping a human decision maker when building a procurement plan under uncertainty. The decision maker knows a set of possible quantities evaluated by possibility levels and can see the consequences of his choices through the inventory evolution. This decision support tool allows the decision maker to make more informed decisions, and also allows to evaluate the risk of the chosen procurement plan. As a perspective, we plan to try to solve this problem as an optimization problem using robust optimization (as minmax or minmaxRegret criteria). The use of these criteria should allow to minimize the maximum possible cost over all the possible scenarios of the gross requirement. Thus, this method should enable to minimize the risk in term of backordering/obsolete inventory in fuzzy uncertain context. Acknowledgments. This study has been performed with the support of Région Midi-Pyrénées and of the University of Toulouse.
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A decision support tool for procurement planning process under uncertainty R. Guillaume*,**, C. Thierry*, B. Grabot** *Université de Toulouse IRIT/UTM, 5, Allées A. Machado, F-31058 Toulouse Cedex 1 France (e-mail: [email protected] , [email protected]). **Université de Toulouse, INPT/LGP/ENIT 47 Av. d'Azereix, BP 1629, F-65016 Tarbes Cedex France (e-mail: [email protected]) Abstract: This communication presents a method to support the customer in the choice of a procurement plan when the gross requirements are ill-known, in a context of collaboration with the supplier. A general model of imperfect parameter representation is suggested, imperfection gathering uncertainty (through various scenarios) and imprecision (through quantities and dates expressed by possibility distribution). A method to compute the possible quantities required to satisfy the gross requirements under the supplier delivering constraints is proposed. From this value, a set of possible supplied quantities is computed to support the decision making of the customer. The decision maker then evaluates the procurement plan with the possible evolution of the inventory. Keywords: supply chain, uncertainty, collaboration, theory of possibility. build the set of all the possible procurement plans. This method is composed of three steps:
1. INTRODUCTION Enterprises are nowadays more and more tightly integrated in supply chains whereas the globalization of the market and the reduction of the product life cycles increases the uncertainty on the customer's demand. Thus, taking into account uncertainty in supply chains is becoming a major issue. If no historical data allowing to build a stochastic model is available, uncertainty is often taken into account in the literature by possibility theory and fuzzy set theory (Dubois and Prade, 1988) (see for instance (Peidro et al., 2009), (Lan et al., 2009), (Aliev et al., 2007)). These methods have for instance been used to model the imprecision on the coefficients of a cost function (Demirli and Yimer, 2006) or the weights of constraints (Mula, 2007). Moreover, in an uncertain environment, a collaborative process between actors of the supply chain (customer, supplier) is a powerful tool to reduce the risk of the bullwhip effect (Galasso et al., 2006). One of the references of collaborative processes in the industry is CPFR (Collaborative Planning forecasting replenishment (Ireland and Crum, 2005)). Within this framework, data such as inventory levels, forecast, etc. are exchanged between the supplier and the customer. In order to take into account more explicitly the uncertainty related to the demand, a method is suggested in this communication aiming at supporting the decision maker (the customer) in building a procurement plan for the supplier. This decision is made under uncertainty on the gross requirements, in collaborative context (the supplier sends to the customer its delivering capacity constraints). To support the decision maker in finding a feasible procurement plan, we
978-3-902661-93-7/11/$20.00 © 2011 IFAC
◊
1) the computation of the required inventory for each period,
◊
2) the computation of the possible procured quantity under the delivering constraints.
◊
3) the evaluation of the proposed procurement plan according to the possible evaluations of the inventory.
This paper is organized as follows: the first section presents a model of imperfect gross requirement. In the second one, a method to compute the required inventory is suggested while in the third section, a method to compute the possible procured quantity for a period is described, together with the framework of a decision support process aiming at building a procurement plan. In the last section, this decision support process is illustrated on an example. 2. IMPRECISE AND UNCERTAIN GROSS REQUIREMENTS Our main hypothesis is that the customer has knowledge on gross requirements including imperfections (uncertainty and imprecision), often empirically modelled through expertise. So, the gross requirements can be given by the customer as a set of scenarios, with an imprecision on the value (e.g. "Not more than twenty parts, but certainly between 15 and 20"). Possibility theory is then used to model the uncertainty and imprecision on the demand in a more analytical way. Without loss of generality, uncertainty is represented as often by a possibility level (definition 1) and the imprecision by trapezoidal possibility distributions (definition 2 and figure 1). All the scenarios are synthesized in a graph (definition 3,
1590
10.3182/20110828-6-IT-1002.01342
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
figure 2) where a path stands for a possible scenario. The method to compute this graph of gross requirement is developed in (Guillaume et al., 2010). Definition 1: A possibility level denotes to what extent the occurrence of event e ∈ E if possible. The possibility level t is an upper bound of the probability that the event e will
To represent a fuzzy number, an interval is used where the extreme values are gradual numbers. This representation enables to apply the interval analysis without using the α-cuts (Fortin et al., 2008). We also use a set of two linear gradual numbers to represent the trapezoidal possibility distribution.
~
Figure 2 illustrates the model for a fuzzy number A . The gradual number is represented by two
~−
appear Pr(e) ≤ Π (e) and ∃e Π (e) = 1 . Definition 2: A possibility distribution πv of v quantifies the plausibility of the information v. πv is a function of S in L such
~−
~−
values A = [ A (0); A represented by 4 values.
as ∀s ∈ S , π v ( s ) ∈ L and ∃s π v ( s ) = 1 with v denoting an ill-
possibility
known value in S, and L the scale of plausibility ([0,1] for the theory of possibility) (Dubois and Prade, 2006).
1
(1)] , so a fuzzy number is
~ A−
~ A+
Possibility
x
1
Fig. 3. Fuzzy number
~ ~ ~ A = [ A− ; A+ ]
Quantity S
The data are denoted as follows: Fig. 1. Trapezoidal distribution of possibility
t: period with t ∈[1,T]
Definition 3: The graph of the gross requirements is a directed arc weighted layer graph G =< V , E > with quantities on node defined as:
V = V1 ∪ ... ∪ VT with T the number of layers ~ ; v~ ;..; v~ } with such as Vi ∩ V j = φ where V = {v 1 2 n ~ v trapezoidal possibility distribution.
Node set
ct: index of fuzzy gross required quantity on period t with ct∈[1,Ct]
~ ~ ~ Grt ,ct = [Gr − t ,ct ; Gr + t ,ct ] : fuzzy gross required quantity on node (t, ct)
wct ,ct+1 ,t : possibility level of the arc linking the node (t, ct)
i
Arc set E
= (v, w) such as v ∈ Vi & w ∈ Vi +1 and the
weights of arc are defined by a matrix with
M = (Π v , w )( v , w)∈E
max (Π v , w , v ∈ Vi & w ∈ Vi +1 ) = 1 (consequence of
with the node (t+1, ct+1)
Lt = [lt− ; lt+ ] : set of possible quantity of deliveries lt on period t 3. COMPUTATION OF THE REQUIRED INVENTORY
possibility theory).
The first step, which consists in computing the required inventory (definition 4), is presented in this section. Start
Definition 4: The required inventory is the quantity required in the inventory to fulfil the gross quantities required under supplier delivering constraints.
End
This method uses an iterative algorithm, from period T to 1. The dependant variables are denoted: Period 1
~ * ~ *− ~ *+ I rt = [ I rt ; I rt ] : fuzzy inventory required for period t
Period 2
Scenario 1 1
which satisfies all the possible gross requirements,
~ ~ − ~ + I r* t = [ I r* t ; I r* t ] : fuzzy inventory required for period t
Scenario 2
which satisfies one of the possible gross requirement,
Scenario 3
Fig. 2. Example of graph for a horizon of 2 period and 3 scenarios 1591
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
~ * ~ *− ~ *+ I rt , ct = [ I rt , ct ; I rt , ct ] : fuzzy inventory required for node (t,ct) which satisfies all the possible gross requirements,
~ ~ − ~ + I r* t , ct = [ I r* t , ct ; I r* t , ct ] : fuzzy inventory required for node (t,ct) which guarantee the respect of one or more possible gross requirements. When the gross requirements are imperfect (uncertain and imprecise), the required inventory is imperfect too. To deal with imprecision, each required quantity becomes a possibility distribution. To deal with uncertainty, only the two extremes cases are computed: the required inventory which satisfies all the possible gross requirements and the required inventory which satisfies one of the possible gross requirements.
To back-propagate the maximum (minimum) required inventory level and keep the representation of the possibility distribution by 4 values (figure 4), an approximate method is used. This method maximises (minimises) two gradual numbers by a linear function which guarantees that all the values of the exact solution are lower (upper) than the approximate solution (for the minimum: equations 4 to 6 and figure 5). In fact, the maximum (minimum) between two values has in the worst case one intersection point (figure 4).
~ A− ~ B− x
The operator used to back-propagate the required inventory depends on the type of required inventory: •
minimum: used for the inventory which satisfies one of the possible gross requirements,
•
maximum: used for the inventory which satisfies all the possible gross requirements
x
Exact result
Approximate result
Fig. 4. Exact/Approximation of minimum
(
)
~ *+ ~ *+ I rt (0) = max I rt ,ct (0) (4) wct ,ct +1 ,t ≠ 0
It is considered that the required inventory for period T+1 is equal to 0. To compute a required inventory (for example: which satisfy all possible gross requirements I~r * ), the
1 − wct ,ct +1 ,t ~ * + ~ *+ ~ *+ ~ *+ × I rt ,ct (1) − I rt (0) I rt ,ct (1) ⇐ I rt ,ct (1) + (5) wct ,ct +1 ,t
optimization problem (equations 1 to 3) is solved for each node (t,ct) (from (T, CT) to (1,1)) and for the four values of ~ *− *− the required inventory (for example: ~ I rt ,c (0) ; I rt ,c (1) ;
∀wct ,ct +1 ,t 0 < wct ,ct +1 ,t < 1
~ *+ I rt , ct (1) ; I~rt ,c *+ (0) ) .
Figure 5 illustrates the equations defining the minimum
t ,ct
t
(
(
)
)
~ *+ ~ + I rt (1) = max I rt*,ct (1) (6) wct , ct +1 ,t ≠ 0
t
t
between to gradual numbers
~ minimize ( I r * t , ct ) (1) s.t.
~ A−
1 0.6
~ ~ ~ I r * t , ct − Grt , ct + l t+ ≥ I r * t +1 (2) ~ I r * t , ct ≥ 0 (3)
~ ~ A − and B − .
~ B− 1 3
7
~ A−
1
~ B−
Equations (7 & 8) 13 x
1 3
11 13
x
Minimum
The same method is applied for the inventory which satisfies one of the possible gross requirements (with I* becoming I*).
1
~
The extreme required inventory (optimist = I r * − t ,ct ; pessimist
~
= I r * + t ,ct ) corresponds to the extreme bound of the set of the
1
~ gross required quantity for the next period ( I r * t +1 ) and the
11
x
~
extreme bound of set the gross requirement ( Grt ,ct ), so we ~ + ~ ~ ~ ~ have I r * − t ,ct for Gr − t , ct and I r *− t +1 ; and I r * + t ,ct for Gr t ,ct
Fig. 5. Illustration of the approximation of the minimum
~
4. CHOOSE A PROCUREMENT PLAN
and I r *+ t +1 . The required inventory is the minimum inventory needed to satisfy a gross requirement, so the minimum required inventory respecting the delivery constrains (whatever the quantity delivered by the supplier is) corresponds to the +
maximum possible quantity procured: l t .
The second and the third steps of the method, respectively the computation of the possible procured quantity and the evaluation of the possible inventory level for a given procured quantity, are described in this section. These two steps are performed consecutively from the period 1 to T. The dependant variables are denoted:
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
~ * ~ *− ~ *+ Pt , ct = [ Pt , ct ; Pt , ct ] : fuzzy procurement quantity for
~ ~ Pt * = max ( Pt ,*ct ) (10) ct
node (t,ct) which satisfies all the possible gross requirements,
~ ~ − ~ + P*t , ct = [ P*t , ct ; P*t , ct ] : fuzzy procurement quantity for node (t,ct) which satisfies one of the possible gross requirements,
~* ~* − ~* + Pt = [ Pt ; Pt ] : fuzzy procurement quantity for period t which satisfies all the possible gross requirements,
~ ~− ~+ P*t = [ P*t ; P*t ] : fuzzy procurement quantity for period t which satisfies one of the possible gross requirement,
~ I t* : fuzzy possible inventory at the end of period t in the case satisfy all the possible gross requirements are satisfied,
~ I*t : fuzzy possible inventory at the end of period t in the
4.2 Computation of fuzzy possible procured quantities From the possible procured quantities, a crisp value is proposed according to a risk level in terms of inventory. For example, the lower bound of the most possible value for the case which satisfies all gross requirements can be proposed as a first suggestion since it is a prudent solution. Then, two resulting inventories are computed (one aiming at satisfying all the possible gross requirements; one aiming at satisfying one of the possible gross requirements), using the minimum operator for the first one, and the maximum operator for second one. The minimum and the maximum are computed like the required inventory (equations 4, 5 and 6). Equations 11 and 12 compute the inventory levels which satisfy all the possible gross requirements (with xt standing for the proposed value by the decision maker).
~ ~ ~ xt + I t *−1 − Grt , ct = I t , ct ~ ~ I t * = min ( I t *,ct ) (12)
case when one of the possible gross requirements is satisfied, xt : value proposed by the decision maker for the period t δ: difference between the required inventory of period t and the possible inventory for the period t. 4.1 Computation of fuzzy possible procured quantities This method is applied for the two extreme cases concerning all the gross requirements or one of the possible gross requirements. The decision maker receives so a fuzzy set for each case. In order to compute the two fuzzy procured quantities, the following optimization model is solved (equations 7 to 9) for the four values (from P~t ,c *− (0) to t ~ *+ Pt , c t (0) ).
ct
Then, the fuzzy possible procured quantities are computed for the next period. From the first proposition of procurement plan (x1 to xT), the decision maker analyses the evolution of the inventory and modifies the procurement plan until it gives him satisfaction. 5. EXAMPLE In this section, we illustrate on an example the method of computation of the procurement plan under uncertainties. 5.1 Data
minimize ( δ ) (7)
The considered planning horizon is 2 periods and the inventory at period 0 is equal to 0. The imperfect gross requirements are represented in Figure 6 and Table 1, and the delivering constraints in Table 2.
s.t.
~
∀ct (11)
~
~
~
δ = Pt ,c *− (0) + I t *−1− (0) − Grt −,c (0) − I rt − (0) (8) t
t
~ *− lt− ≤ Pt , ct (0) ≤ lt+ (9)
ϭ
ϭ
The required inventory for period 0 is different from the real inventory at the beginning of period 0, which is known. So we are looking for the procured quantity which satisfies the required inventory for the next period (period 1). We associate a possibility level to the possible delivering quantity, corresponding to the possibility level that the quantity will be required. We compute then the fuzzy quantity delivered for the period. In the case of the procurement plan which satisfies all the possible gross requirements, we merge the nodes with the max operator, in the other case with the min operator (equation 10: procurement plan which satisfies all the possible gross requirements)
^ƚĂƌƚ
ϭ ~ Gr2,1
~ Gr1,1
Ϭ͘ϴ
ŶĚ ϭ
~ Gr2, 2
Fig. 6. Imperfect gross requirement Table 1. Gross requirement t, ct
~ Grt −,ct (0)
~ Grt −, ct (1)
~ Grt +, ct (1)
~ Grt +, ct (0)
2,1
12
17
17
25
2,2
15
16
20
22
1,1
6
7
7
8
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Table 2. Delivery constraints Period 1
1 0.8
Period 2
[l ; l ] = [5;17] [l ; l ] = [5;15]
Delivery quantity
1 1
+ 1
1 2
+ 2
0 1
2
1 0.8
From these data, the required inventory is computed.
0 2
3.75
10
5.2 Computation of the required inventory 0
It is considered that the required inventory of period 3 is (0;0;0;0). Then, the required inventory for the node 1 of period 2 is computed as follows. For the last period, the required inventory which satisfies all the gross requirements is equal to the required inventory which satisfies one of the possible gross requirements
~
( Ir
*
2 ,1
~ ~ ~ = I r *2,1 & I r * 2, 2 = I r *2, 2 ).
2
5
7 10
Fig. 7. Maximum Required inventory of period 1 In the same way, but using the min operator, we compute the required inventory which satisfies one of the possible gross requirements (Figure 8). The result of the computation of the required inventory is illustrated in Figure 9.
The lower bound is then computed:
1 0.8
~ minimize ( I r * − 2,1 ) s.t.
~ ~ ~ I r * − 2,1 − Gr − 2,1 + l 2+ ≥ I r * − 3 ~ I r * − 2,1 ≥ 0
0 1
2
1 0.8
In this case, the equation (5) becomes:
0 1.25 2
~ ~ I r * − 2,1 (0) − 12 + 15 ≥ 0 ⇒ I r * − 2,1 (0) = 0 ~ ~ I r * − 2,1 (1) − 17 + 15 ≥ 0 ⇒ I r * − 2,1 (1) = 2
0
The upper bound is:
2
5
7
7 10
Fig. 8. Minimum Required inventory of period 1
~ ~ I r * + 2,1 (0) − 25 + 15 ≥ 0 ⇒ I r * + 2,1 (0) = 10 ~ ~ I r * + 2,1 (1) − 19 + 15 ≥ 0 ⇒ I r * + 2,1 (1) = 2 ~ ~ I r * 2,1 = I r *2,1 = (0;2;2;10) and in the same way, the ~* ~ node (2,2) is computed : I r 2 , 2 = I r *2 , 2 = (0;1;5;7) .
0 0 2
So,
3.75
0 1.25 2
From these results, the required inventory of the node 1 of the period 1 is computed. The back-propagation of the maximum required by the inventory of the period 2 is used for the case which satisfy all the gross requirements (otherwise, the minimum). We apply the equation (4), then the equation (5) and the equation (6). Figure 7 illustrates the calculation.
(
~ *+ ~ *+ ~ *+ I r2 (0) = max I r2 ,1 (0); I r2 ,1 (0)
)
= max(10;7 ) = 10 1 − 0.8 ~ *+ I r2, 2 (1) = 5 + × (5 − 10) = 3.75 0.8
(
~ *+ ~ *+ ~ *+ I r2 (1) = max I r2,1 (1); I r2,1 (1) = max(2; 3.75) = 3.75
)
2
10
10
7 0 1
5
7
Fig. 9. Required inventory 5.3 Computation of fuzzy possible procured quantities First, we compute the fuzzy procurement plan which satisfies all the gross requirements. We apply the equations 11, 12 and 13 at the period 1:
~
δ = P1,1*− (0) + 0 − 6 − 0 ~ *− 5 ≤ P1,1 (0) ≤ 17 ~ *− ~* − so P1,1 (0) = P1 (0) = 6 1594
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
In the same way, we have:
~* + P1 (0) = 17 .
~* − ~* + P1 (1) = 9 P1 (1) = 10.75
5.4 Computation of fuzzy possible procured quantities A prudent solution is proposed: the lower bound of the most possible value for the case which satisfies all gross requirements (value=9). On the left of Figure 10, a set of fuzzy procured quantities is proposed. On the right, the inventory level resulting from the proposition is represented, together with the required inventory level for the next period. Then, the decision maker can modify the first crisp value and the right side to evaluate the new proposition. In the example, he proposes the quantity 11, which leads to a less risked solution. Fuzzy procurement
Inventory level
All GR
6
9 10.75
17
2
3.75
10
1 GR
6 8 9 Proposition 1 :9
15
1.25 2
7
Proposition 2 :11 Inventory = (1 ;2 ;2 ;3) Inventory = (3;4 ;4 ;5)
Fig. 10. Dashboard of the decision maker for period 1 6. CONCLUSIONS
7. REFERENCES Ireland, R. and Crum, C. (2005). Supply chain Collaboration: how to implement CPFR® and other best collaborative practices, Integrated Business Management series and APICS Dubois, D. and Prade, H. (1988). Possibility Theory, Plenum Press New York. Aliev, R.A., Fazlollahi, B., Guirimov B.G. and Aliev, R.R. (2007). Fuzzy-genetic approach to aggregate production– distribution planning in supply chain management. Information Sciences, 177 (20), 4241–4255. Guillaume, R., Thierry, C. and Grabot, B. (2010). Integration of ill-known requirements with dependencies into a gross requirement plan, 8ème ENIM IFAC Conférence Internationale de Modélisation et Simulation, May 10-12 Hammamet, Tunisia. Galasso, F., Mercé, C. and Grabot, B. (2009). Decision Support for Supply Chain Planning under uncertainty. International Journal of Systems Science, 39 (7), 667675. Dubois, D. and Prade, H. (2006). Représentations formelles de l’incertain et de l’imprécis. In : Hermès – Lavoisier, eds. Concepts et méthodes pour l’aide à la décision 1, 11-165. Mula, J., Polera, R. and Garcia-Sabater, J.P. (2007). Material Requirement Planning with fuzzy constraints and fuzzy coefficients, Fuzzy Sets and Systems, 158 (7), 783 – 793. Peidro, D., et al., (2009). Fuzzy optimization for supply chain planning under supply, demand and process uncertainties. Fuzzy Sets and Systems, 160 (18), 26402657. Lan, Y.F., Liu, Y-K. and Sun, G-J. (2009). Modeling fuzzy multi-period production planning and sourcing problem with credibility service levels, Journal of Computational and Applied Mathematics, 231 (1), 208-221. Fortin, J., Dubois, D. and Fargier, H. (2008). Gradual Numbers and Their Application to Fuzzy Interval Analysis, Fuzzy Systems, IEEE Transactions on , 16(2), pp.388-402, doi: 10.1109/TFUZZ.2006.890680
In this paper is proposed a decision support tool for helping a human decision maker when building a procurement plan under uncertainty. The decision maker knows a set of possible quantities evaluated by possibility levels and can see the consequences of his choices through the inventory evolution. This decision support tool allows the decision maker to make more informed decisions, and also allows to evaluate the risk of the chosen procurement plan. As a perspective, we plan to try to solve this problem as an optimization problem using robust optimization (as minmax or minmaxRegret criteria). The use of these criteria should allow to minimize the maximum possible cost over all the possible scenarios of the gross requirement. Thus, this method should enable to minimize the risk in term of backordering/obsolete inventory in fuzzy uncertain context. Acknowledgments. This study has been performed with the support of Région Midi-Pyrénées and of the University of Toulouse.
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